Problem 32 - Algebra and Trigonometry

Question

Find all solutions of the system of equations. \(\left\\{\begin{aligned} x^{2}+2 y^{2} &=2 \\ 2 x^{2}-3 y &=15 \end{aligned}\right.\)

Step-by-Step Solution

Answer

There are no real solutions.

Step 1: Analyze the given system of equations

We have two equations: \(x^2 + 2y^2 = 2\) and \(2x^2 - 3y = 15\). The first equation is a quadratic form in terms of \(x\) and \(y\), while the second equation is also a quadratic form involving \(x^2\) and a linear term involving \(y\).

Step 2: Express \(x^2\) from the first equation

From the first equation \(x^2 + 2y^2 = 2\), solve for \(x^2\) to get \(x^2 = 2 - 2y^2\).

Step 3: Substitute \(x^2\) in the second equation

Substitute \(x^2 = 2 - 2y^2\) into the second equation \(2x^2 - 3y = 15\). This gives \(2(2 - 2y^2) - 3y = 15\).

Step 4: Simplify and solve for \(y\)

Expand the expression: \(4 - 4y^2 - 3y = 15\). Rearrange to form a quadratic equation in terms of \(y\): \(-4y^2 - 3y - 11 = 0\). Multiply through by -1 to get \(4y^2 + 3y + 11 = 0\).

Step 5: Check for real solutions of \(y\)

Calculate the discriminant \(D = b^2 - 4ac\) for the equation \(4y^2 + 3y + 11 = 0\). Here, \(a = 4\), \(b = 3\), and \(c = 11\). Thus, \(D = 3^2 - 4 \times 4 \times 11 = 9 - 176 = -167\). The discriminant is negative, which means there are no real solutions for \(y\). As a result, there are no real solutions for the system.

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation that involve terms up to the second degree. They take the standard form \( ax^2 + bx + c = 0 \) where:

• \( a \), \( b \), and \( c \) are constants,
• \( x \) is the variable to solve for,
• and \( a \) cannot be zero.

Quadratic equations can describe a wide range of phenomena, from physics to finance. They form a parabola when graphed, opening either upwards or downwards depending on the sign of \( a \). For our system, the first equation, \( x^2 + 2y^2 = 2 \), is quadratic in both \( x \) and \( y \). Understanding the properties of quadratics, such as their symmetrical nature and vertex, can be key in solving complex systems of equations.

Substitution Method

The substitution method is a strategy to solve systems of equations. It involves expressing one variable in terms of another and substituting this expression into another equation. This reduces the number of variables and simplifies the system. Here's why it's useful:

• You simplify and solve the system step-by-step, making it easier to handle, especially for nonlinear equations.
• By reducing variables, you often end up with a simpler, single-variable equation to solve.

In our exercise, we expressed \( x^2 \) from the first equation as \( x^2 = 2 - 2y^2 \) and substituted this into the second equation, helping us focus on finding \( y \) first.

No Real Solutions

When working with quadratic equations, sometimes the solution set has no real numbers that satisfy the equations. This happens when the equations describe conditions that cannot be possible simultaneously. This concept is crucial in understanding the limitations of a mathematical model or problem. If the calculated result signals no real solutions, this usually means:

• The equations provided model a scenario with no intersection points in reality.
• You might be dealing with an incorrect or unsolvable problem under real-number conditions.

For the given system, after simplifying the equations and calculating the quadratic discriminant, it was determined that there were no real solutions for \( y \), thus no real intersection or solution for the system at large.

Discriminant in Quadratics

The discriminant is a key component in solving quadratic equations, found in the expression \( b^2 - 4ac \) of the standard quadratic equation \( ax^2 + bx + c = 0 \). It tells us about the nature of the roots:

• If the discriminant \( D > 0 \), the quadratic has two distinct real roots.
• If \( D = 0 \), there is one real, repeated root.
• If \( D < 0 \), there are no real roots, only complex ones.

In the exercise, we found the discriminant for the equation \( 4y^2 + 3y + 11 = 0 \) to be -167, indicating no real solutions. This negative value distinctly shows that the system of equations has no points of intersection where both conditions are true in the real number space.